Hardness results and approximation algorithms for some problems on graphs

This thesis has two parts. In the first part, we study some graph covering problems with a non-local covering rule that allows a "remote" node to be covered by repeatedly applying the covering rule. In the second part, we provide some results on the packing of Steiner trees. In the Propagation problem we are given a graph $G$ and the goal is to find a minimum-sized set of nodes $S$ that covers all of the nodes, where a node $v$ is covered if (1) $v$ is in $S$, or (2) $v$ has a neighbor $u$ such that $u$ and all of its neighbors except $v$ are covered. Rule (2) is called the propagation rule, and it is applied iteratively. Throughout, we use $n$ to denote the number of nodes in the input graph. We prove that the path-width parameter is a lower bound for the optimal value. We show that the Propagation problem is NP-hard in planar weighted graphs. We prove that it is NP-hard to approximate the optimal value to within a factor of $2^{\log^{1-\epsilon}{n}}$ in weighted (general) graphs. The second problem that we study is the Power Dominating Set problem. This problem has two covering rules. The first rule is the same as the domination rule as in the Dominating Set problem, and the second rule is the same propagation rule as in the Propagation problem. We show that it is hard to approximate the optimal value to within a factor of $2^{\log^{1-\epsilon}{n}}$ in general graphs. We design and analyze an approximation algorithm with a performance guarantee of $O(\sqrt{n})$ on planar graphs. We formulate a common generalization of the above two problems called the General Propagation problem. We reformulate this general problem as an orientation problem, and based on this reformulation we design a dynamic programming algorithm. The algorithm runs in linear time when the graph has tree-width $O(1)$. Motivated by applications, we introduce a restricted version of the problem that we call the $\ell$-round General Propagation problem. We give a PTAS for the $\ell$-round General Propagation problem on planar graphs, for small values of $\ell$. Our dynamic programming algorithms and the PTAS can be extended to other problems in networks with similar propagation rules. As an example we discuss the extension of our results to the Target Set Selection problem in the threshold model of the diffusion processes. In the second part of the thesis, we focus on the Steiner Tree Packing problem. In this problem, we are given a graph $G$ and a subset of terminal nodes $R\subseteq V(G)$. The goal in this problem is to find a maximum cardinality set of disjoint trees that each spans $R$, that is, each of the trees should contain all terminal nodes. In the edge-disjoint version of this problem, the trees have to be edge disjoint. In the element-disjoint version, the trees have to be node disjoint on non-terminal nodes and edge-disjoint on edges adjacent to terminals. We show that both problems are NP-hard when there are only $3$ terminals. Our main focus is on planar instances of these problems. We show that the edge-disjoint version of the problem is NP-hard even in planar graphs with $3$ terminals on the same face of the embedding. Next, we design an algorithm that achieves an approximation guarantee of $\frac{1}{2}-\frac{1}{k}$, given a planar graph that is $k$ element-connected on the terminals; in fact, given such a graph the algorithm returns $k/2-1$ element-disjoint Steiner trees. Using this algorithm we get an approximation algorithm with guarantee of (almost) $4$ for the edge-disjoint version of the problem in planar graphs. We also show that the natural LP relaxation of the edge-disjoint Steiner Tree Packing problem has an integrality ratio of $2-\epsilon$ in planar graphs

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

The complexity of the Pk partition problem and related problems in bipartite graphs

In this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems

The complexity of the Pk partition problem and related problems in bipartite graphs

International audienceIn this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems

On Approximability of Steiner Tree in ℓp\ell_p-metrics

In the Continuous Steiner Tree problem (CST), we are given as input a set of points (called terminals) in a metric space and ask for the minimum-cost tree connecting them. Additional points (called Steiner points) from the metric space can be introduced as nodes in the solution. In the Discrete Steiner Tree problem (DST), we are given in addition to the terminals, a set of facilities, and any solution tree connecting the terminals can only contain the Steiner points from this set of facilities. Trevisan [SICOMP'00] showed that CST and DST are APX-hard when the input lies in the $\ell_1$-metric (and Hamming metric). Chleb\'ik and Chleb\'ikov\'a [TCS'08] showed that DST is NP-hard to approximate to factor of $96/95\approx 1.01$ in the graph metric (and consequently $\ell_\infty$-metric). Prior to this work, it was unclear if CST and DST are APX-hard in essentially every other popular metric! In this work, we prove that DST is APX-hard in every $\ell_p$-metric. We also prove that CST is APX-hard in the $\ell_{\infty}$-metric. Finally, we relate CST and DST, showing a general reduction from CST to DST in $\ell_p$-metrics. As an immediate consequence, this yields a $1.39$-approximation polynomial time algorithm for CST in $\ell_p$-metrics.Comment: Abstract shortened due to arxiv's requirement

Lossy Kernelization

In this paper we propose a new framework for analyzing the performance of preprocessing algorithms. Our framework builds on the notion of kernelization from parameterized complexity. However, as opposed to the original notion of kernelization, our definitions combine well with approximation algorithms and heuristics. The key new definition is that of a polynomial size $\alpha$-approximate kernel. Loosely speaking, a polynomial size $\alpha$-approximate kernel is a polynomial time pre-processing algorithm that takes as input an instance $(I,k)$ to a parameterized problem, and outputs another instance $(I',k')$ to the same problem, such that $|I'|+k' \leq k^{O(1)}$. Additionally, for every $c \geq 1$, a $c$-approximate solution $s'$ to the pre-processed instance $(I',k')$ can be turned in polynomial time into a $(c \cdot \alpha)$-approximate solution $s$ to the original instance $(I,k)$. Our main technical contribution are $\alpha$-approximate kernels of polynomial size for three problems, namely Connected Vertex Cover, Disjoint Cycle Packing and Disjoint Factors. These problems are known not to admit any polynomial size kernels unless $NP \subseteq coNP/poly$. Our approximate kernels simultaneously beat both the lower bounds on the (normal) kernel size, and the hardness of approximation lower bounds for all three problems. On the negative side we prove that Longest Path parameterized by the length of the path and Set Cover parameterized by the universe size do not admit even an $\alpha$-approximate kernel of polynomial size, for any $\alpha \geq 1$, unless $NP \subseteq coNP/poly$. In order to prove this lower bound we need to combine in a non-trivial way the techniques used for showing kernelization lower bounds with the methods for showing hardness of approximationComment: 58 pages. Version 2 contain new results: PSAKS for Cycle Packing and approximate kernel lower bounds for Set Cover and Hitting Set parameterized by universe siz